Realizing curves as geodesics of positively curved metrics

It is a well known fact (see here for instance), that under reasonable conditions any curve on a smooth manifold can be realized as a geodesic for some given connection.

The natural construction when $$\gamma$$ is closed, for example, consists in to realize a Riemannian metric in the form: $$g = fg_1 + hg_2,$$ where $$g_2$$ is a flat Riemannian metric and $$\{f,h\}$$ is a unit partition. In this construction, $$\gamma$$ is a geodesic for the metric $$g_2$$.

I was wondering: is there a manner to make such analogous construction preserving positive curvature? Or at least, for some planes?

Imagine you have a Riemannian manifold $$(M,\tilde g)$$ such that $$\tilde g$$ has positive sectional curvature and a smooth curve $$c$$ on $$M$$. Is it possible to obtain a Riemannian metric $$g$$ (possible related to $$\tilde g$$) on $$M$$ in which $$c$$ is a geodesic and the curvature of $$g$$ is also positive?

If we try obtain a Riemannian metric with positive sectional curvature following the construction on the second paragraph, then I imagine a natural way to proceed is using the arbitrarily of $$g_1$$ to impose further conditions on the unity partition. I elaborate:

$$K_{hg_2}$$ by the formulae of conformal change only depends on the Hessian of the function $$h$$. Since $$h$$ has compact support it has directions where the Hessian can be negative and this can make the curvature of $$g$$ be negative. But we also have the curvature of $$fg_1$$. It seems naîve, but couldn't we just search for: we ask that $$g_1$$ be such that when the Hessian of $$h$$ is negative (for example when $$h$$ is decreasing) then since $$\{f,h\}$$ is a unity partition, $$f$$ is increasing, then it has positive Hessian, we ask that the curvature of $$g_1$$ is positive on such directions, and where $$f$$ has negative Hessian, then we ask $$K_{g_1}$$ ti be negative. I have many suspicious that this doesn't work although I cannot justify.

I appreciate any hep